A reduction theory for non-self-adjoint operator algebras
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- by E. A. Azoff, C. K. Fong and F. Gilfeather PDF
- Trans. Amer. Math. Soc. 224 (1976), 351-366 Request permission
Abstract:
It is shown that every strongly closed algebra of operators acting on a separable Hilbert space can be expressed as a direct integral of irreducible algebras. In particular, every reductive algebra is the direct integral of transitive algebras. This decomposition is used to study the relationship between the transitive and reductive algebra problems. The final section of the paper shows how to view direct integrals of algebras as measurable algebra-valued functions.References
- William B. Arveson, A density theorem for operator algebras, Duke Math. J. 34 (1967), 635–647. MR 221293
- Edward A. Azoff, $K$-reflexivity in finite dimensional spaces, Duke Math. J. 40 (1973), 821–830. MR 331081
- Edward A. Azoff and Frank Gilfeather, Measurable choice and the invariant subspace problem, Bull. Amer. Math. Soc. 80 (1974), 893–895. MR 361831, DOI 10.1090/S0002-9904-1974-13560-3
- J. A. Dyer, E. A. Pedersen, and P. Porcelli, An equivalent formulation of the invariant subspace conjecture, Bull. Amer. Math. Soc. 78 (1972), 1020–1023. MR 306947, DOI 10.1090/S0002-9904-1972-13090-8 J. A. Dyer and P. Porcelli, Concerning the invariant subspace problem, Notices Amer. Math. Soc. 17 (1970), 788. Abstract #677-47-4.
- Edward G. Effros, The Borel space of von Neumann algebras on a separable Hilbert space, Pacific J. Math. 15 (1965), 1153–1164. MR 185456
- C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72. MR 367142, DOI 10.4064/fm-87-1-53-72
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 397–403 (English, with Russian summary). MR 188994 O. Maréchal, Topologie et structure borélienne sur l’ensemble des algèbres de von Neumann, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A847-A850. MR 47 #5611.
- F. I. Mautner, Unitary representations of locally compact groups. I, Ann. of Math. (2) 51 (1950), 1–25. MR 32650, DOI 10.2307/1969494
- John von Neumann, On rings of operators. Reduction theory, Ann. of Math. (2) 50 (1949), 401–485. MR 29101, DOI 10.2307/1969463
- Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682
- Heydar Radjavi and Peter Rosenthal, A sufficient condition that an operator algebra be self-adjoint, Canadian J. Math. 23 (1971), 588–597. MR 417802, DOI 10.4153/CJM-1971-066-7
- D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517. MR 192365
- J. T. Schwartz, $W^{\ast }$-algebras, Gordon and Breach Science Publishers, New York-London-Paris, 1967. MR 0232221
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 224 (1976), 351-366
- MSC: Primary 46L15; Secondary 47A15
- DOI: https://doi.org/10.1090/S0002-9947-1976-0448109-1
- MathSciNet review: 0448109